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5 REPRESENTATIONS OF ALGEBRAS WITH INVOLUTION 345
20. Let A be a Banach algebra with unit element <?, and let x*-+x* be a (not necessarily
continuous) involution on A. Show that if ||jc|| ^p(x), then A is a star algebra.
5. REPRESENTATIONS OF ALGEBRAS WITH INVOLUTION
Let A be an algebra with involution (not necessarily endowed with a
norm, and not necessarily having a unit element), and let H be a Hilbert
space. A representation^ of A in H is a homomorphism s\-~+ U(s) of A into
the algebra St? (H) of continuous endomorphisms of H, such that
(15.5.1) U(s*) = (U(s))*.
This implies in particular that if s is self-adjoint, then so is U(s). If A has a
unit element e, we require in addition that
The representation Uis said to be faithful if the homomorphism jh-* U(s) is
infective, that is if the relation U(s) • x = 0 for all x e H implies that s = 0.
Let H, H' be two Hilbert spaces. A representation s\-+ U(s) of A in H and
a representation s\-*U'(s) of A in H' are said to be equivalent if there
exists a Hilbert space isomorphism T : H -> H' such that U'(s) = TC/^)!1"1
for all s G A.
When H = H', the Hilbert space automorphisms T of H are precisely the
unitary elements of the star algebra JSf(H). For T must be invertible and
satisfy (T - x \ T - y) = (x \ y) for all x, y in H, so that (T*T * x\y) = (x\y) and
therefore T*T= 1H, hence T* = T'1 ; the converse is immediate.
Let H1? H2 be two Hilbert spaces, s\-+ U^s) a representation of A in Hl5
and 5-F-> U2(s) a representation of A in H2 . Let H be the Hilbert sum of
H! and H2 , so that Hj and H2 are identified with supplementary subspaces
of H. If x = Xi + x2 and y = yl + y2 are two elements of H (where
xt , yi in H£ for i = 1, 2), then (6.4)
If we put U(s) - (x1 + x2) = Ui(s) - xt 4- U2(s) - x2 , it is immediately verified
that U(s) E ^f(H) for each s e A, and that jh-> U(s) is a representation of A,
called the Hilbert sum of the given representations.
t Strictly speaking, a unitary representation. Since we shall not consider other types of
representations, we shall suppress the word "unitary," by abuse of language,.